2025/10/08 by Jonathan Leake, Shayan Oveis Gharan, Leake, Jonathan +1
Mathematics · #Advanced Operator Algebra Research #Combinatorics (math.CO) #Computational Complexity (cs.CC) #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #Geometric Analysis and Curvature Flows #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2510.06549
openalex publication_date 2025/10/08 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
Let μ be a probability distribution on a multi-state spin system on a set V of sites; equivalently, a d-partite simplicial complex with distribution μ on maximal faces. For any pair of vertices u,v∈ V, define the pairwise spectral influence Iu,v as follows. Let σ be a choice of spins sw∈ Sw for every w∈ V∖\u,v\, and construct a matrix in ℝ(Su∪ Sv)× (Su∪ Sv) where for any su∈ Su, sv∈ Sv, the (usu,vsv)-entry is the probability that sv is the spin of v conditioned on su being the spin of u and on σ. Then Iu,v is the maximal second eigenvalue of this matrix, over all choices of spins for all w∈ V∖\u,v\. Equivalently, Iu,v is the maximum local spectral expansion of links of codimension 2 that include a spin for every w ∈ V ∖ \u,v\. We show that if the largest eigenvalue of the pairwise spectral influence matrix with entries Iu,v is bounded away from 1, i.e. λmax(I)≤ 1-ε (and X is connected), then the Glauber dynamics mixes rapidly and generate samples from μ. This improves/generalizes the classical Dobrushin's influence matrix as the Iu,v lower-bounds the classical influence of u→ v. As an application, we prove that the Glauber dynamics mixes rapidly up to (approximately) the phase transition for the multi-state hardcore model--a widely studied model in telecommunication networks and statistical physics (generalizing the hardcore model) introduced by Mazel and Suhov. As a by-product of our results, we also prove improved/almost optimal trickle-down theorems for partite simplicial complexes. Our proof builds on the trickle-down theorems via C-Lorentzian polynomials machinery recently developed by the authors and Lindberg.